Proposition: Stress constraints alone do not imply an Einstein equation

Symmetry, conservation, and vanishing trace of a tensor do not, without a metric–matter equation, imply $G_{\mu \nu }=8\pi G_{\mathrm{N}}{T}_{\mu \nu }$.

Proof. On any Lorentzian geometry take $T_{\mu \nu }=0$. It is symmetric, conserved, and trace free. Choose a geometry whose Einstein tensor is nonzero, for example a four-dimensional constant-curvature geometry with $R_{\mu \nu }=3H^2{g}_{\mu \nu }$, $H\ne 0$. Then

The zero-cosmological-constant Einstein equation with that $T_{\mu \nu }$ therefore fails.